Why it matters on the HSPT
Fractions are the single most tested idea in the two math sections. They appear directly in Mathematics as arithmetic and word problems, and they hide inside Quantitative Skills, where a comparison item asks which of three fractions is largest and a manipulation item says one fourth of what number is 15. Roughly one question in seven across both sections needs a fraction move, and no calculator is allowed, so the moves have to be automatic.
Everything on this page uses numbers small enough to do in your head or in the margin. That is on purpose. The test rarely uses large denominators, because it is testing whether you know the rules, not whether you can grind arithmetic.
The rules
A fraction is a division. Three fourths means 3 divided by 4. The top number, the numerator, counts pieces; the bottom number, the denominator, says how many pieces make one whole. That is why 3/4 is less than 1 and 5/4 is more than 1.
Equivalent fractions. Multiplying or dividing the top and bottom by the same number does not change the value. 2/3 equals 4/6 equals 10/15. To simplify, divide top and bottom by the largest number that goes into both: 12/18 becomes 2/3 after dividing both by 6. Always simplify your answer, because the test's choices are always in lowest terms.
Comparing fractions. Three ways, fastest first.
- Same denominator: bigger numerator wins. 5/8 is more than 3/8.
- Same numerator: bigger denominator loses, because the pieces are smaller. 3/5 is more than 3/7.
- Anything else: cross-multiply. To compare 4/7 and 5/9, multiply 4 times 9 to get 36 and 5 times 7 to get 35. The 36 belongs to 4/7, so 4/7 is larger. A shortcut first: if one fraction is above one half and the other below, you are done. 4/7 is above one half because 4 is more than half of 7.
Adding and subtracting. The denominators must match. If they do, add or subtract the numerators and keep the denominator: 3/8 plus 1/8 is 4/8, which simplifies to 1/2. If they do not match, rewrite each fraction over a common denominator first, usually the smallest number both denominators divide into. For 1/4 plus 1/6, the common denominator is 12: 3/12 plus 2/12 is 5/12.
Multiplying. Multiply straight across, top times top and bottom times bottom. 2/3 times 3/5 is 6/15, which is 2/5. Faster: cancel before you multiply. The 3 on top and the 3 on the bottom cancel, leaving 2/5 immediately. Of means multiply: two thirds of 30 is 2/3 times 30, which is 20.
Dividing. Keep the first fraction, flip the second, multiply. 3/4 divided by 1/2 is 3/4 times 2/1, which is 6/4, which is 3/2 or one and a half. Notice that dividing by a fraction smaller than 1 makes the answer bigger, which is a good sanity check.
Mixed numbers. Two and a half is 5/2. To convert, multiply the whole number by the denominator and add the numerator: 2 times 2 plus 1 is 5, over 2. Convert to an improper fraction before multiplying or dividing; for adding, you can work the whole numbers and fractions separately.
Worked examples
Example 1
Which fraction is equivalent to 18/24?
- A.2/3
- B.3/4
- C.4/6
- D.6/9
Show answer
Answer: B: 3/4
The largest number that divides both 18 and 24 is 6. Dividing both by 6 gives 3/4. Choice A, 2/3, comes from dividing by 9 on top and 8 on the bottom, which is not allowed; both must be divided by the same number.
Example 2
Which is the largest: 3/5, 5/8, or 7/12?
- A.3/5
- B.5/8
- C.7/12
- D.They are equal
Show answer
Answer: B: 5/8
All three are above one half, so compare in pairs. 3/5 against 5/8: 3 times 8 is 24, 5 times 5 is 25, so 5/8 wins. 5/8 against 7/12: 5 times 12 is 60, 7 times 8 is 56, so 5/8 wins again. Choice B.
Example 3
1/3 + 1/4 =
- A.2/7
- B.1/12
- C.7/12
- D.2/12
Show answer
Answer: C: 7/12
The denominators differ, so find the common one: 12. One third is 4/12 and one fourth is 3/12. Their sum is 7/12. Choice A, 2/7, is the trap for adding tops and bottoms separately, which is never correct.
Example 4
5/6 − 1/2 =
- A.4/4
- B.1/3
- C.2/3
- D.4/6
Show answer
Answer: B: 1/3
Rewrite 1/2 as 3/6. Then 5/6 minus 3/6 is 2/6, which simplifies to 1/3. Choice D, 4/6, is the trap for subtracting only the numerators without matching the denominators first.
Example 5
3/4 × 8/9 =
- A.2/3
- B.24/36
- C.11/13
- D.3/2
Show answer
Answer: A: 2/3
Cancel first. The 4 and the 8 share a factor of 4, leaving 1 and 2. The 3 and the 9 share a factor of 3, leaving 1 and 3. Now multiply 1/1 times 2/3, which is 2/3. Choice B, 24/36, is the same value but not simplified, and the test never lists an unsimplified answer as correct.
Example 6
2/5 ÷ 4/15 =
- A.8/75
- B.3/2
- C.2/3
- D.6/5
Show answer
Answer: B: 3/2
Keep, flip, multiply: 2/5 times 15/4. Cancel the 5 with the 15 to get 3, and the 2 with the 4 to get 2. That leaves 3/2. Choice A, 8/75, is what you get by multiplying without flipping.
Example 7
Two thirds of a number is 18. What is the number?
- A.12
- B.27
- C.36
- D.54
Show answer
Answer: B: 27
Two thirds of the number is 18, so one third is 9, and the whole number is three times 9, which is 27. Check: two thirds of 27 is 18. Choice A, 12, is two thirds of 18, the forward computation the question does not ask for.
Common mistakes
- Adding across. 1/2 plus 1/3 is not 2/5. Denominators never add; they must match first.
- Forgetting to simplify. 6/8 is correct but will not be among the choices; 3/4 will.
- Flipping the wrong fraction when dividing. Only the second fraction flips. Keep the first as it is.
- Multiplying denominators when you could cancel. 4/9 times 3/8 is much easier as 1/3 times 1/2 after canceling than as 12/72.
- Comparing by looking. 5/9 and 6/11 are too close to eyeball. Cross-multiply and be sure.
Practice
These moves appear in arithmetic, in non-geometric comparison, and in number manipulation. Practice all three, and expect to see fractions inside word problems as well.